Sequences & series worksheet with answers

Free printable Sequences & series worksheet: 6 problems from easy to exam level, with an answer key. Print it or save it as a PDF.

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1.Easy
kak​123456789101112

Find the sum S22 of the first 22 terms of the arithmetic sequence 7, 3, −1, −5, …

a₁ = 7d = −4n = 22
S22 =
2.Easy
kak​13579111315ε = 0.01

The geometric sequence aₙ = 3·0.75ⁿ⁻¹ decreases. Find the first index n for which aₙ < 0.01.

a₁ = 3r = 0.75ε = 0.01
n =
3.Medium
AΣ 2ᵏ/k²k = 1 … ∞BΣ 1/k³k = 1 … ∞CΣ k³/3ᵏk = 1 … ∞DΣ (−1)ᵏ/√kk = 1 … ∞

Exactly one of the four series diverges. Click it.

A: Σ 2ᵏ/k²B: Σ 1/k³C: Σ k³/3ᵏD: Σ (−1)ᵏ/√k
  1. 1A: Σ 2ᵏ/k²
  2. 2B: Σ 1/k³
  3. 3C: Σ k³/3ᵏ
  4. 4D: Σ (−1)ᵏ/√k
4.Medium
xx₀ = 0??R

Find the radius of convergence R of the power series Σk=1∞ k!·xᵏ/kᵏ.

Σ k!·xᵏ/kᵏx₀ = 0
R =
5.Hard

f(x) = eˣ is approximated at x = 0.5 by its Maclaurin polynomial of degree n. With the Lagrange bound M·|x|ⁿ⁺¹/(n + 1)! and M = 2, find the smallest degree n that guarantees an error below 1 × 10⁻⁵.

f(x) = eˣx = 0.5M = 2ε = 1 × 10⁻⁵
n =
6.Hard

Recognise the series and find its sum: S = Σk=1∞ (−1)k+1·0.2ᵏ/k.

ak = (−1)k+1·0.2ᵏ/k
S =

Answer key

  1. 1. S22 = −770
  2. 2. n = 21
  3. 3. A: Σ 2ᵏ/k²
  4. 4. R = 2.718
  5. 5. n = 6
  6. 6. S = 0.1823

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